# Optimal Algorithms for Online Convex Optimization with Adversarial Constraints Source: [Optimal Algorithms for Online Convex Optimization with Adversarial Constraints]() Collection index: [llms.txt]() Data updated: 2026-01-01T12:39:25.142Z Authors: Abhishek Sinha, Rahul Vaze Published: 2023-10-29T09:55:41.000Z ## Abstract A well-studied generalization of the standard online convex optimization (OCO) framework is constrained online convex optimization (COCO). In COCO, on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to design an online learning policy that simultaneously achieves a small regret while ensuring a small cumulative constraint violation (CCV) against an adaptive adversary interacting over a horizon of length $T$. A long-standing open question in COCO is whether an online policy can simultaneously achieve $O(\\sqrt{T})$ regret and $\\tilde{O}(\\sqrt{T})$ CCV without any restrictive assumptions. For the first time, we answer this in the affirmative and show that a simple first-order policy can simultaneously achieve these bounds. Furthermore, in the case of strongly convex cost and convex constraint functions, the regret guarantee can be improved to $O(\\log T)$ while keeping the CCV bound the same as above. We establish these results by effectively combining adaptive OCO policies as a blackbox with Lyapunov optimization - a classic tool from control theory. Surprisingly, the analysis is short and elegant. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()